# A Sage implementation of the Virasoro algebra and its representations

Hello everyone,

A few years ago I tried to implement the Virasoro algebra on Sage, but due to having a lot of other things to do, I didn't really get anywhere. Now I'd like to try again in earnest. But due to not having a lot of experience with programming, I'm having a hard time reading the Sage documentation. So I was hoping people could give me some pointers to get me going.

First some math: The Virasoro algebra is a complex infinite dimensional Lie algebra with generators $L_n, n\in \mathbb{Z}$ and bracket $[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}(m^3-m)\frac{C}{12}$, where $\delta$ is the Kronecker delta and $c$ is the central element of the Virasoro algebra (aka central charge).

A Verma module $M(h,c)$ of the Virasoro algebra is generated by a highest weight vector $v(h,c)$ such that $L_0v(h,c)=hv(h,c)$, $Cv(h,c)=cv(h,c)$ and $L_nv(h,c)=0,\ n>0$. The remaining generators act freely up to the relations coming from the Virasoro algebra itself. Therefore, a basis of $M(h,c)$ is given by $L_{-n_1}L_{-n_2}\cdots L_{-n_m}v(h,c)$, where $n_1\geq n_2\geq \cdots \geq n_n\geq1,\ m\geq0$, i.e. it is parametrised by all partitions of integers.

Here is what I've learned from talking to people on this forum a few years ago:
Since the Virasoro algebra and its Verma modules are parametrised by integers and partitions of integers, it seems natural to define them using CombinatorialFreeModule, e.g.
`Vir=CombinatorialFreeModule(QQbar,Integer(), prefix='L')`

(this does not include the central element $C$). Unfortunately I'm clueless as to how to define the Lie bracket, orderings of generators and an action on the Verma module.

The list below is a detailed list of things I wish to implement. If someone could help me with one or two of those, I think I should be able to figure out the rest by example.

- Define a bracket on the generators that extends linearly to the whole Virasoro algebra.
- Define an ordering of generators such that a multiplication of generators can be defined, i.e. the universal enveloping algebra. The ordering is: If $m < n$ then $L_m \cdot L_n$ just stays $L_m\cdot L_n$ but $L_n\cdot L_m=[L_n,L_M]+L_m\cdot L_n$. All the better if this product could overload the multiplication symbol *.
- Define an action of Virasoro generators on basis elements of $M(h,c)$ and extend this action linearly to all of $M(h,c)$.
- Extend the action of Virasoro generators on $M(h,c)$ to products of Virasoro generators acting on $M(h,c)$.

Thanks in advance for any advice.